PSFC/JA-08-16
Limitations of gyrokinetics on transport time scales
Parra, F.I. and Catto, P.J
February 2008
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge MA 02139 USA
This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER54109. Reproduction, translation, publication, use and disposal, in whole or in part, by or
for the United States government is permitted.
Accepted for publication in Plasma Phys. Control. Fusion, Vol. 50 (June 2008)
Limitations of Gyrokinetics on Transport Time
Scales
Felix I Parra and Peter J Catto
Plasma Science and Fusion Center, MIT, Cambridge, MA 02139, USA
E-mail: fparra@mit.edu, catto@psfc.mit.edu
Abstract. We present a new recursive procedure to find a full f electrostatic
gyrokinetic equation correct to first order in an expansion of gyroradius over magnetic
field characteristic length. The procedure provides new insights into the limitations
of the gyrokinetic quasineutrality equation. We find that the ion distribution
function must be known at least to second order in gyroradius over characteristic
length to calculate the long wavelength components of the electrostatic potential selfconsistently. Moreover, using the example of a steady-state θ-pinch, we prove that
the quasineutrality equation fails to provide the axisymmetric piece of the potential
even with a distribution function correct to second order. We also show that second
order accuracy is enough if a more convenient moment equation is used instead of the
quasineutrality equation. These results indicate that the gyrokinetic quasineutrality
equation is not the most effective procedure to find the electrostatic potential if the
long wavelength components are to be retained in the analysis.
PACS numbers: 52.25.Dg, 52.25.Fi, 52.30.Gz, 52.35.Ra
Submitted to: Plasma Phys. Control. Fusion
Limitations of Gyrokinetics on Transport Time Scales
2
1. Introduction
Nonlinear gyrokinetics have proven extremely useful for studying drift turbulence in the
tokamak core. In the last decade, continuum flux-tube δf models [1, 2] have been used to
satisfactorily calculate the short wavelength spectrum of turbulence and the associated
transport. These δf codes assume that the ion and electron distribution functions are
Maxwellian at long wavelengths, and only calculate the turbulent, short wavelength δf
portion of the distribution function to obtain the turbulent particle and heat transport.
In recent years there has been an increasing interest in extending these turbulence
calculations to longer wavelengths and transport timescales and obtaining self-consistent
radial profiles for tokamaks. The electric field is of special importance since the poloidal
zonal flow [3, 4, 5, 6] induced by its radial structure can act to control the saturated
amplitude of turbulence. Calculating the electric field is an incompletely solved problem
even when turbulence is not considered. The axisymmetric radial electric field has only
been recently found in the Pfirsch-Schlüter regime [7, 8, 9], and there has been some
incomplete work on the banana regime for high aspect ratio tokamaks [10, 11]. Most
results are obtained in the high flow limit [12, 13, 14, 15] that will not be considered
here. Consequently, a gyrokinetic model appropriate for transport time scales has to
face the unsolved challenge of providing the axisymmetric radial electric field, as well
as retaining all relevant turbulence effects including its interaction with neoclassical
transport (and other serious difficulties associated with large computations).
We focus on the subtleties associated with determining the long wavelength portion
of the radial electric field. However, we also retain the shorter wavelength zonal flow and
turbulent behavior in our electrostatic gyrokinetic model formulated to first order in a
gyroradius over characteristic length expansion (current gyrokinetic codes usually work
to this order only). The formalism used to find the nonlinear gyrokinetic variables is
similar to the technique presented in [16, 17] for linear gyrokinetics [18, 19, 20]. Care is
taken to insure that for long wavelengths, the result recovers the gyrophase dependent
piece of the distribution function to second order, as already found in drift-kinetics
[21, 22].
In δf gyrokinetics, a modified quasineutrality equation has been traditionally used
to solve for the electrostatic potential [23, 24]. The difference between the density of
gyrocenters and the real ion density, due to the effect of the short wavelength components
of the electric field on the gyromotion, is adjusted to ensure quasineutrality. In the
process, the short wavelength components of the electric field are determined. The
calculated turbulent fluxes are reasonably close to the experimental values [25, 26]. This
methodology differs strongly from the procedures used in drift-kinetics [21, 27], where
some form of ∇ · J = 0 is employed to find the electrostatic potential. We carefully
examine the possibility of extending the gyrokinetic approach to longer wavelengths in
order to determine the axisymmetric electric field. We find that the gyrokinetic equation
is not known to high enough order to give a meaningful result. This conclusion is of
great importance, because GYRO [28] can be run in a global mode and several groups
Limitations of Gyrokinetics on Transport Time Scales
3
[29, 30, 31] have already began to develop codes that solve for the full distribution
function, and they intend to use the gyrokinetic quasineutrality equation to find the
potential, including the long wavelength pieces. An issue we address is whether their
results at long wavelengths will be flawed because of the limitations of the traditional
gyrokinetic approach.
We are aware that gyrokinetic equations of the same or even higher order than ours
(with geometrical restrictions) have been derived by different authors [32, 33, 34, 35]
using Hamiltonian approaches. Our method is an alternative approach that allows us
to determine the missing ingredients in the gyrokinetic distribution function and, just
as importantly, the limitations of the usual gyrokinetic quasineutrality equation. These
missing pieces are the main reason the gyrokinetic quasineutrality equation should not
be used to find the axisymmetric electric field.
The rest of this article is organized as follows. In section 2 we present the orderings
and formalism used to find the gyrokinetic variables, and we obtain the gyrokinetic
equation to first order in a gyroradius over characteristic length expansion. The
formalism in section 2 is required in section 3 to derive quasineutrality in gyrokinetic
form and to highlight its shortcomings at long wavelengths. Details of the derivations
are relegated to the appendices. Section 4 illustrates the problems of the gyrokinetic
quasineutrality equation by applying the gyrokinetic approach to the simplified geometry
of the θ-pinch. Finally, in section 5 we discuss our findings.
2. Gyrokinetic variables and Fokker–Planck equation
This section is devoted to the derivation of convenient non-linear gyrokinetic variables.
Only electrostatic gyrokinetics is considered. We assume that the magnetic field does
not change in time and that it has slow spatial variation. Since the magnetic field is
constant in time, the electric field can be expressed as a function of the electrostatic
potential, φ, by E = −∇φ. The slow spatial variation of the magnetic field implies the
existence of a small parameter δ = ρ/L ¿ 1, with L = |∇(ln B)|−1 the characteristic
length for the magnetic field and ρ = M cvi /ZeB the ion gyroradius, where
pB and
B = |B| are the magnetic field and the magnitude of the magnetic field, vi = 2Ti /M
is the ion thermal velocity, Z and M are the charge number and the mass of the species
of interest, and e and c are the electron charge and the speed of light.
2.1. Orderings
The characteristic frequency of the processes of interest is assumed to be the drift wave
frequency ω ∼ ω∗ ∼ k⊥ ρvi /L. To treat arbitrary collisionality, the ion collision frequency
is assumed to be of the order of the transit time of ions, ν ∼ vi /L.
We consider the drift ordering, where the E × B drift is of order δvi . Therefore, the
electrostatic potential is O(T /e), where T ∼ Ti ∼ Te , and the electric field is of order
E = −∇φ = O(T /eL).
(1)
Limitations of Gyrokinetics on Transport Time Scales
4
Similarly, the spatial gradient of the distribution functions is assumed to be
∇f = O(fM /L),
(2)
where fM is the zeroth order distribution function. For estimates, we will assume that
the zeroth order distribution function is a slowly varying Maxwellian, with the density
and temperature in the Maxwellian having characteristic lengths of variation Ln,T ∼ L
much larger than the ion gyroradius. Most of our results are valid for any slowly varying
zeroth order distribution function, but to make estimates it is convenient to work with
a Maxwellian. Moreover, it is also a reasonable assumption since we are primarily
interested in the core plasma in tokamaks and other well confined plasmas.
Our gyrokinetic description must resolve both neoclassical (k⊥ L ∼ 1) and turbulent
(k⊥ ρ ∼ 1) spatial scales. Hence, we will allow components of φ and f with short
perpendicular wavelengths, k⊥ L À 1. Such components have a slow variation along
the magnetic field: n̂ · ∇ ∼ 1/L, with n̂ = B/B. The size of the short wavelength
components of the electric field, φk , is determined by the ordering of the E × B drift.
According to (1), the gradient of φk is |∇φk | = k⊥ φk ∼ T /eL. This relationship sets a
maximum size for φk ,
eφk /T ∼ (k⊥ L)−1 >
∼ δ,
(3)
where k⊥ L >
∼ 1. For k⊥ L ∼ 1, the potential is of the order of the temperature, but as k⊥
grows, the size of the corresponding potential component decreases. For k⊥ ρ ∼ 1, the
potential is given by eφk /T ∼ δ ¿ 1. We are interested in the components that have
wavelengths on the order of or longer than the ion gyroradius, which means that the
electrostatic potential φ must be determined to O(δT /e) at least.
To treat a possible adiabatic or Maxwell-Boltzmann response, we will order the
short wavelength component of the distribution function, fk , consistent with the
electrostatic potential by taking
fk /fM ∼ (k⊥ L)−1 >
∼ δ.
(4)
As with the potential, the components with k⊥ ρ ∼ 1 are O(δfM ) so that ∇fk ∼ k⊥ δfM ∼
fM /L. Hence, the distribution function must be solved to O(δfM ) or higher.
Both the potential and the distribution function may be viewed as having a slowly
spatially varying piece (representing the average value in the plasma) plus some rapid
oscillations of small amplitude. The zonal flows, for example, will be included in the
small piece if their characteristic wavelength is comparable to the gyroradius, but their
amplitude may be larger for larger wavelengths. An advantage of this view point is that
the rapid spatial potential fluctuations seen by a particle in its gyromotion are small
compared to the average value of the potential amplitude. Similarly, the distribution
function of the gyrocenters is equal, to zeroth order, to the distribution function of
the particles. The difference, coming from the rapidly oscillating pieces, is small in
our ordering. Notice that the δf codes [1, 28] explicitly adopt this treatment for the
components of φ and f that satisfy k⊥ ρ ∼ 1, and, as in this work, they order them as
O(δ).
Limitations of Gyrokinetics on Transport Time Scales
5
2.2. Gyrokinetic variables
We begin by defining the Vlasov operator in the usual r, v variables for an electrostatic
electric field as the following total derivative
d/dt = ∂/∂t + v · ∇ + [−(Ze/M )∇φ + Ωv × n̂] · ∇v ,
(5)
where Ω = ZeB/M c is the gyrofrequency. The Fokker–Planck equation is then simply
df /dt = C{f },
(6)
where C is the relevant Fokker–Planck collision operator. Our goal is to change the
Fokker–Planck equation to gyrokinetic variables in such a way that all the required
gyrophase information is retained to higher order than standard gyrokinetic treatments
[2]. Many of the algebraic details are relegated to Appendices A to C. Appendices A
and B give the complete derivation of the first and second order gyrokinetic variables.
To obtain the conservative form of the gyrokinetic equation the Jacobian is required.
Details are presented in Appendix C. Also, our gyrokinetic variables allow us to find
the gyroviscosity for long wavelengths. The calculation is shown in Appendix D and the
result is the same as in [22]. We could write a higher order gyrokinetic Fokker-Planck
equation based on these variables, but it would be tedious and the result is not needed.
Therefore, only the variables are given to higher order.
The nonlinear gyrokinetic variables to be employed are the guiding center location
R, the kinetic energy E, the magnetic moment µ, and the gyrophase ϕ. These variables
will be defined to higher order than is customary by employing an extension of the
procedure presented in [16] for high frequency gyrokinetics. The general idea is to
construct the gyrokinetic variables to higher order by adding in δ corrections such that
the total derivative of a generic gyrokinetic variable Q is gyrophase independent to the
desired order, and we may safely employ
dQ/dt ' hdQ/dti,
(7)
where the gyrophase average h. . .i is performed holding R, E, µ and t fixed. The
gyrokinetic variable Q is expanded in powers of δ,
Q = Q0 + Q1 + Q2 + . . . ,
(8)
where Q0 is the lowest order gyrokinetic variable (kinetic energy, magnetic moment,
etc.), and Q1 = O(δQ0 ), Q2 = O(δ 2 Q0 )... are the order δ, δ 2 ... corrections. The first
correction Q1 is constructed so that dQ/dt = hdQ/dti + O(δ 2 ΩQ), while the second
correction Q2 is evaluated such that dQ/dt = hdQ/dti + O(δ 3 ΩQ). In principle this
process can be continued indefinitely. Any Qk can be found once the functions Qm , for
m = 1, 2, . . . , k − 1, are known. All the functions Qm are constructed so that
¿
À
d
d
dQ
' (Q0 + . . . + Qk−1 ) =
(Q0 + . . . + Qk−1 ) + O(δ k ΩQ0 ). (9)
dt
dt
dt
Adding Qk means adding dQk /dt to (9). To lowest order, dQk /dt ' −Ω ∂Qk /∂ϕ, which
to the requisite order leads to an equation for Qk ,
¿
À
d
∂Qk
d
dQ
' (Q0 + . . . + Qk−1 ) − Ω
=
(Q0 + . . . + Qk−1 ) ,
(10)
dt
dt
∂ϕ
dt
Limitations of Gyrokinetics on Transport Time Scales
6
where h∂Qk /∂ϕi = 0 is employed. Using (10), Qk = O(δ k Q0 ) is found to be periodic in
gyrophase and given by
·
¿
À¸
Z
1 ϕ 0 d
d
Qk =
dϕ
(Q0 + . . . + Qk−1 ) −
(Q0 + . . . + Qk−1 ) . (11)
Ω
dt
dt
More explicitly, through the first two orders, Q1 and Q2 are determined to be
µ
¿
˦
Z
1 ϕ 0 dQ0
dQ0
Q1 =
−
dϕ
(12)
Ω
dt
dt
and
1
Q2 =
Ω
Z
·
ϕ
0
dϕ
d
(Q0 + Q1 ) −
dt
¿
À¸
d
(Q0 + Q1 ) .
dt
(13)
By adding Q1 and Q2 , the total derivative of the gyrokinetic variable Q =
Q0 + Q1 + Q2 is
dQ/dt = hd(Q0 + Q1 )/dti + O(δ 3 ΩQ0 ).
(14)
In the reminder of this subsection, we present the gyrokinetic variables that result
from this process. We begin with the kinetic energy expanded as
E = E0 + E1 + E2 + . . . ,
(15)
where E0 = v 2 /2, E1 = O(δvi2 ) and E2 = O(δ 2 vi2 ). We construct E1 and E2 such that
the energy derivative is gyrophase independent to order δ,
dE/dt = hdE/dti + O(δ 2 vi3 /L).
(16)
The explicit details are presented in Appendices A and B. We find
e
E1 = Zeφ/M
(17)
e
E2 = (c/B)(∂ Φ/∂t),
(18)
and
e are functions related to the electrostatic potential. They depend on
where φ, φe and Φ
the new gyrokinetic variables. Their definitions are
I
1
φ(R, E, µ, t) = hφi =
dϕ φ(r(R, E, µ, ϕ, t), t),
(19)
2π
e
φ(R,
E, µ, ϕ, t) = φ(r(R, E, µ, ϕ, t), t) − φ(R, E, µ, t),
and
Z
e
Φ(R,
E, µ, ϕ, t) =
ϕ
e
E, µ, ϕ0 , t),
dϕ0 φ(R,
(20)
(21)
e = 0. These are the same definitions used by Dubin [32].
such that hΦi
It is important to comment on the size of these functions. Both φ and φ are of the
same order as the temperature for long wavelengths, but small for short wavelengths.
However, φe is always small as it accounts for the variation in the electrostatic potential
that a particle sees as it moves in its gyromotion. Of course, since the potential is small
Limitations of Gyrokinetics on Transport Time Scales
7
for short wavelengths, the variation observed by the particle is also small. For long
wavelengths, even though the potential is comparable to the temperature, the particle
motion is small compared to the wavelength, and the variations that it sees in its motions
e small as
are small. Therefore, φe ∼ δT /e for all wavelengths in our ordering, making Φ
well.
The Vlasov operator acting on E is shown in Appendix B to give
¿ À
µ 3¶
µ 3¶
¤
dE
Ze £
dE
v
2 vi
=
+O δ
=−
v || n̂(R) + vd · ∇R φ + O δ 2 i ,(22)
dt
dt
L
M
L
where vd is the total drift velocity, composed of E × B drift and magnetic drift vM
vd = −(c/B)∇R φ × n̂ + vM ,
(23)
vM = (v 2|| /Ω)n̂ × (n̂ · ∇R n̂) + (µ/Ω)n̂ × ∇R B.
(24)
with vM
In the preceding equations, v || is the gyrocenter parallel velocity defined by
v 2|| /2 + µB(R) = E.
(25)
Note that in (22), (23), (24) and (25), all the terms are given as a function of the new
gyrokinetic variables, R, E and µ.
The gyrokinetic gyrophase is obtained in a similar way as the energy by defining
ϕ = ϕ0 + ϕ1 + . . . ,
(26)
with ϕ0 the original gyrophase. The details are again in Appendix A. The most
important result is that dϕ/dt is gyrophase independent to order δ, that is,
dϕ/dt = hdϕ/dti + O(δ 2 Ω) = −Ω + O(δ 2 Ω),
(27)
where Ω ' Ω to lowest order and Ω is constructed to be gyrophase independent through
order δ. The final result for Ω is given by (A.12).
For the guiding center position we define
R = R0 + R1 + R2 + . . . ,
(28)
where R0 = r, |R1 | = O(ρ) and |R2 | = O(δρ).
Proceeding with R in a similar manner as for E and ϕ we find the usual result [18]
R1 = Ω−1 v × n̂.
(29)
To next order we obtain
µ ¶
¶
µ
¶¸
·µ
v||
n̂
1
1
1
·
R2 =
v|| n̂ + v⊥ (v × n̂) + (v × n̂) v|| n̂ + v⊥
+ 2 v⊥ · ∇n̂
×∇
Ω
4
4
Ω
Ω
½
¾
n̂
1
c
e × n̂,
+ 2 v|| v⊥ · (n̂ · ∇n̂) + [v⊥ v⊥ − (v × n̂)(v × n̂)] : ∇n̂ −
∇R Φ
(30)
Ω
8
BΩ
which is the same as [16] except for the nonlinear term given last. Our vector conventions
↔
↔
↔
↔
are xy:M= y· M ·x and xy ×· M= x × (y· M). The Vlassov operator acting on R then
gives
dR/dt = hdR/dti + O(δ 2 vi ) = v || n̂(R) + vd + O(δ 2 vi ),
(31)
Limitations of Gyrokinetics on Transport Time Scales
8
where vd is given by (23).
The gyrokinetic magnetic moment variable is dealt with somewhat differently since
we want to construct it to remain an adiabatic invariant order by order. The condition
for the magnetic moment is not only that its derivative must be gyrophase independent,
but that hdµ/dti must vanish order by order, giving
dµ/dt = O(δ 2 vi3 /BL) ' 0,
(32)
for µ to remain an adiabatic invariant. We define
µ = µ 0 + µ1 + µ2 + . . . ,
(33)
2
/2B is the usual lowest order result, µ1 = O(δµ0 ) and µ2 = O(δ 2 µ0 ).
where µ0 = v⊥
For µ to remain an adiabatic invariant, µ1 and µ2 must contain gyrophase independent
contributions such that hdµ/dti = 0 to the requisite order. Solving for µ1 as outlined in
Appendix A gives
2
v||
v|| v⊥
Ze e 1
n̂ · ∇ × n̂. (34)
φ − v⊥ · vM −
[v⊥ (v × n̂) + (v × n̂)v⊥ ] : ∇n̂ −
MB
B
4BΩ
2BΩ
2
To keep µ an adiabatic invariant, hµ1 i = −(v|| v⊥
/2BΩ)(n̂ · ∇ × n̂) 6= 0. The calculation
of µ2 has been omitted for brevity. It turns out that the function µ2 is not needed in
what follows since we do not allow strong µ variation in f .
µ1 =
2.3. Fokker–Planck equation
The Fokker–Planck equation (6) becomes
∂f /∂t + Ṙ · ∇R f + Ė ∂f /∂E + µ̇ ∂f /∂µ + ϕ̇ ∂f /∂ϕ = C{f }.
(35)
when written in gyrokinetic variables, where Q̇ = dQ/dt, and Q is any of the gyrokinetic
variables. The gyroaverage of this equation is
∂f /∂t + Ṙ · ∇R f + Ė ∂f /∂E = hC{f }i,
(36)
where f (R, E, µ, t) = hf i. Here, we have used that R, E and µ are defined such
that their time derivatives are gyrophase independent to the orders given by (22), (31)
and (32). Therefore, in (36) we have neglected pieces that are O(fM δ 2 vi /L). We also
have neglected the term hϕ̇ ∂f /∂ϕi = O(feδvi /L), where fe = f − f is the gyrophase
dependent piece of the distribution function. We will prove in the next paragraph
that fe is O(fM δν/Ω), making all the neglected terms smaller than fM δ 2 vi /L, and the
distribution function gyrophase independent to first order, f ' f . Notice that, due to
the missing pieces, we can only obtain contributions to the distribution function that
are O(δfM ), as well as all terms with k⊥ ρ ∼ 1.
The explicit equation for the gyrophase dependent part of the distribution function
is obtained by subtracting from the full Fokker–Planck equation (35) its gyroaverage,
giving to lowest order
−Ω ∂ fe/∂ϕ = C{f } − hC{f }i.
(37)
Limitations of Gyrokinetics on Transport Time Scales
9
Therefore, the collisional term is the one that sets the size of fe. In many cases, the
distribution function is a Maxwellian to zeroth order. This means that the collision
operator vanishes to zeroth order, C{f } = O(δνfM ), giving C{f }−hC{f }i = O(δνfM ).
As a result, fe is
µ
¶
Z
1 ϕ 0
δν
e
f '−
dϕ (C{f } − hC{f }i) = O
fM ,
(38)
Ω
Ω
where ν/Ω ¿ 1.
Using the values of dE/dt from (22) and dR/dt from (31), and suppressing the
overbar by using f ' f , the equation for f in gyrokinetic variables is
∂f /∂t + [v || n̂(R) + vd ] · [∇R f − (Ze/M )∇R φ ∂f /∂E] = hC{f }i,
(39)
where φ is the function defined in (19), and f is gyrophase independent.
The gyrokinetic equation can be also written in conservative form. To do so, the
Jacobian of the gyrokinetic transformation is needed. Conservation of particles in phase
space requires the Jacobian of the transformation, J = ∂(r, v)/∂(R, E, µ, ϕ), to satisfy
∂J/∂t + ∇R · (ṘJ) + ∂(ĖJ)/∂E + ∂(µ̇J)/∂µ + ∂(ϕ̇J)/∂ϕ = 0.
(40)
(This is the equality ∇ · ṙ + ∇v · v̇ = 0 written in gyrokinetic variables). Employing this
property, equation (35) can be written in conservative form by multiplying it by J to
obtain
³
´
´
∂
∂ ³
∂
∂
(Jf ) + ∇R · ṘJf +
ĖJf +
(µ̇Jf ) +
(ϕ̇Jf ) = JC{f }.
(41)
∂t
∂E
∂µ
∂ϕ
The gyroaverage of this equation is
∂(Jf )/∂t + ∇R · (ṘJf ) + ∂(ĖJf )/∂E = JhC{f }i.
(42)
We have taken into account that the Jacobian J is independent of ϕ to the order of
interest, as can be seen by using (40). The equation for the gyrophase dependent part of
the Jacobian is obtained by subtracting from (40) its gyroaverage. Notice that J − hJi
depends on the differences Ṙ − hṘi, Ė − hĖi..., and those differences are small by
definition of the gyrokinetic variables. The gyrophase-dependent part of the Jacobian is
estimated to be J − hJi = O(δ 2 B/vi ). Finally, we substitute dR/dt and dE/dt in (42)
to get
¾
½
∂
∂
Ze
(Jf ) + ∇R · {Jf [v || n̂(R) + vd ]} −
[v || n̂(R) + vd ] · ∇R φ = JhC{f }i.(43)
Jf
∂t
∂E
M
The calculation of the Jacobian is described in Appendix C. The final result is
J = ∂(r, v)/∂(R, E, µ, ϕ) = B(R)/v || + (M c/Ze)(n̂ · ∇R × n̂).
(44)
In Appendix C we also prove that J satisfies the gyroaverage of (40).
Similar gyrokinetic equations to (39) and (43) can be found for the gyrokinetic
variables R, v || and µ, where v || is defined by (25). From (22), (25), (31) and dµ/dt ' 0
we find
v̇ || = −[n̂(R) + (v || /Ω)n̂ × (n̂ · ∇R n̂)] · ∇R [µB(R) + Zeφ/M ],
(45)
Limitations of Gyrokinetics on Transport Time Scales
10
which gives the gyrokinetic equation
∂f /∂t + [v || n̂(R) + vd ] · ∇R f + v̇ || ∂f /∂v || = hC{f }i.
(46)
This gyrokinetic equation can be written in conservative form by noticing that the new
Jacobian is given by
Jv|| = ∂(r, v)/∂(R, v || , µ, ϕ) = B(R) + (M cv || /Ze)(n̂ · ∇R × n̂).
(47)
Using the new Jacobian, we can write
∂(Jv|| f )/∂t+∇R ·{Jv|| f [v || n̂(R)+vd ]}+∂(Jv|| f v̇ || )/∂v || = Jv|| hC{f }i.(48)
3. Quasineutrality equation for a gyrokinetic distribution function
The distribution function fi in Poisson’s equation,
· Z
¸
2
3
−∇ φ = 4πe Z d v fi (R, E, µ, t) − ne (r, t) ,
(49)
is obtained from (39) or (43). Therefore, it is known as a function of the gyrokinetic
variables. The distribution function can be rewritten more conveniently as a function
of r + Ω−1 v × n̂ and v by Taylor expanding. However, it is important to remember that
there are missing pieces of order δ 2 in the distribution function since terms of this order
must be neglected to derive (39) and (43). Thus, the expansion can only be carried out
to the order where the distribution function is totally known, resulting in
fi (R, E, µ, t) = fi (Rg , E0 , µ0 , t) + E1 ∂fi /∂E0 + µ1 ∂fi /∂µ0 + O(δ 2 fM ).
(50)
Notice that Rg ≡ r + Ω−1 v × n̂ cannot be Taylor-expanded if we allow k⊥ ρ ∼ 1.
In addition to this Taylor expansion, we also take into account that the turbulent
wavelengths we are interested in are usually much larger than the Debye length.
Thus, the term in the left side of Poisson’s equation may be neglected. The resulting
quasineutrality equation reduces to
µ
¶
Z
Z 2e
∂fi
1 ∂fi
3 e
d vφ
+
' −Z N̂i (r, t) + ne (r, t).
(51)
M
∂E0 B ∂µ0
where ne is the electron density, the function N̂i is the ion guiding center number density
when the effect of the electrostatic potential is extracted,
and terms of order O(δ 2 ne )
p
and O(ne λ2D /Lρ) have been neglected, where λD = T /4πe2 ne is the Debye length and
k⊥ ρ ∼ 1. It is convenient to write N̂i as
Z
Z
1
3
N̂i (r, t) = d v fi (Rg , E0 , µ0 , t) + n̂ · ∇ × n̂ d3 v v|| fi .
(52)
Ω
The last integral comes from the term µ1 ∂fi /∂µ0 upon integrating by parts and is
negligible when fi is a stationary Maxwellian to lowest order. In the higher order
integrals involving φe in (51), the function fi must be taken only to lowest order.
Moreover, according to our ordering, the corrections arising from using Rg instead of
r are small in these two terms because, even though we allow small wavelengths, the
amplitude of the fluctuations with small wavelengths is assumed to be of the next order.
Limitations of Gyrokinetics on Transport Time Scales
11
Therefore, in the higher order integrals, only the long wavelength distribution function
depending on Rg ' r need be retained.
Equation (51) may be used to calculate φ for wavelengths of the order of the
gyroradius, including zonal flow, as is normally done in δf turbulence codes such as
GS2 [1] or GYRO [28]. However, the equation is not useful for long wavelengths. In the
limit k⊥ L ∼ 1, the average of φe holding r, v|| and v⊥ fixed becomes the same order as
terms already neglected since
e + Ω−1 (v × n̂), E0 , µ0 , ϕ0 , t) = φ(r,
e E0 , µ0 , ϕ0 , t) + O(δ 2 T /e).
φ(r
(53)
As a result, the terms on the left side of (51) vanish to the order the equation has been
derived, leaving
Z N̂i (r, t) = ne (r, t)
(54)
as the quasineutrality equation. This equation does not depend at all on φ. Therefore,
we cannot solve for the correct φ at long wavelengths. Either a moment description or
a more accurate gyrokinetic treatment are required to solve for φ.
It is possible to obtain a higher order long wavelength quasineutrality equation
if the ion distribution function is assumed to be known to high enough order. The
resulting equation in the long wavelength limit, assuming a Maxwellian distribution to
lowest order, is
µ
¶
Zcni
ZM c2 ni
∇·
∇⊥ φ −
|∇⊥ φ|2 = ne − Z N̂i ,
(55)
BΩ
2Ti B 2
where N̂i is defined to higher order,
Z
³
´
↔
v||
∇∇pi
3
N̂i (r, t) = d v fi (r, E0 , µ0 , t) 1 + n̂ · ∇ × n̂ + ( I −n̂n̂) :
.
(56)
Ω
2M Ω2
Even though this equation is correct, it is only useful if we are able to evaluate
the missing O(δ 2 ) pieces in fi that are of the same order as the left side in (55).
Equations (39) or (43) miss these pieces. The derivation of (55) is shown in Appendix
E.
4. Gyrokinetic solution of the θ-pinch at long wavelengths
In the θ-pinch, the magnetic field is given by B = B(r)n̂, where here n̂ is a constant
unit vector in the axial direction, and r is the radial coordinate in cylindrical geometry.
For long wavelengths, we can find the gyrokinetic equation to order δ 2 . The simplified
geometry of the magnetic field yields more manageable expressions for the gyrokinetic
variables, i.e., µ1 and R2 become
e B − (v 2 /2B 2 Ω)(v × n̂) · ∇B
µ1 = Zeφ/M
⊥
(57)
R2 = (4BΩ2 )−1 [v⊥ v⊥ − (v × n̂)(v × n̂)] · ∇B,
(58)
and
Limitations of Gyrokinetics on Transport Time Scales
12
e × n̂ has been neglected because we assume that k⊥ L ∼ 1.
where the term (c/BΩ)∇R Φ
Using R2 , we calculate the gyroaverage of Ṙ,
hṘi ' hv|| in̂ + h(µ0 /Ω)n̂ × ∇B − (c/B)∇φ × n̂ + v⊥ · ∇R2 − (Ze/M )∇φ · ∇v R2 i, (59)
where we have used the fact that in a θ-pinch n̂ · ∇B = 0 to write v · ∇R2 = v⊥ · ∇R2 .
The gyroaverages are performed by employing the long wavelength approximation
∇φ ' ∇R φ − Ω−1 (v × n̂) · ∇R φ and the relation hv⊥ v⊥ v⊥ i = 0, to get
hṘi ' hv|| in̂ + [µ/Ω(R)]n̂ × ∇R B − [c/B(R)]∇R φ × n̂.
(60)
The gyroaverage of Ė is found by using (B.3) to write
e
e
Ė = −(Ze/M )(v · ∇φ − dφ/dt)
' −(Ze/M )(Ṙ · ∇R φ + µ̇ ∂φ/∂µ − ∂ φ/∂t),
(61)
where we employ that ∂/∂t ∼ δvi /L for long wavelengths and ∂φ/∂E = O(δ 3 M/e)
in the θ-pinch. Considering that hµ̇i = O(δ 2 vi3 /BL) and ∂φ/∂µ = O(δM B/e), the
gyroaverage of (61) is calculated to be
hĖi ' −(Ze/M )hṘi · ∇R φ.
(62)
Thus, the gyrokinetic equation to order O(δ 2 ) is
∂fi /∂t + hṘi · [∇R fi − (Ze/M )∇R φ ∂fi /∂E] = hC{fi }i.
(63)
We have neglected the derivative ∂fi /∂µ because the distribution function is Maxwellian
to zeroth order and hµ̇i is already small by definition of µ. For an axisymmetric steady
state solution, the terms on the left side of (63) vanish, the second term because the
gyrocenter parallel and perpendicular drifts, hṘi, remain in surfaces of constant fi and
φ (therefore hv|| i need not be evaluated to second order). Thus, equation (63) becomes
hC{fi }i = 0. Such an equation can be solved for a simplified collision operator. We
use a Krook operator, C{fi } = −ν(fi − fM ), with constant collision frequency ν and a
shifted Maxwellian,
fM = ni (M/2πTi )3/2 exp[−M (v − Vi )2 /2Ti ],
(64)
where ni , Ti and Vi are functions of the position r. We assume that the parallel average
velocity, Vi|| = n̂ · Vi , is zero and we order Vi as O(δvi ) to obtain
fM ' fM 0 [1 + M v⊥ · Vi /Ti + M 2 (v⊥ · Vi )2 /2Ti2 − M Vi2 /2Ti ],
(65)
fM 0 = ni (M/2πTi )3/2 exp(−M v 2 /2Ti ).
(66)
with
With the Krook operator, the gyrokinetic solution is
·
³ cn
´
M c2
x2⊥
x2⊥
i
2
2
fi = hfM i = f M 1 −
∇·
∇⊥ φ +
(2
−
x
)|∇
φ|
−
∇2⊥ pi
⊥
⊥
2
2
ni
BΩ
2Ti B
2ni M Ω
µ
µ
¶
¶
2
2
x⊥
35
5
2x⊥
+
− 7x2 + x4 |∇⊥ Ti |2 +
− x2 ∇ ⊥ B · ∇ ⊥ T i
2
2
2Ti M Ω
4
M BΩ 2
¶
¸
µ
µ
¶
x2⊥
5
5
M Vi2 2
c
2
2
2
2
∇⊥ φ · ∇ ⊥ Ti
− x⊥ − x +
x −
(x⊥ − 1) ,(67)
+
∇⊥ Ti +
Ti BΩ
2
2M Ω2
2
2T
Limitations of Gyrokinetics on Transport Time Scales
13
2
where x2 = M v 2 /2Ti ' M E/Ti , x2⊥ = M v⊥
/2Ti ' M µB/Ti and
f M = ni (R)[M/2πTi (R)]3/2 exp[−M E/Ti (R)].
(68)
To obtain this equation we have employed
Vi = (ni M Ω)−1 n̂ × ∇pi − (c/B)∇φ × n̂.
(69)
The distribution function in (67) has been calculated by using a gyrokinetic equation
that is correct to order δ 2 for both the left side and the gyroaveraged collision operator.
Using the definitions R = r + R1 + R2 , E = E0 + E1 + E2 and µ = µ0 + µ1 , and the
gyroaverage collisional piece fei given in (38), we can find the distribution function fi in
r, v variables to order O(δ 2 fM 0 ). As a check, the same solution has been also obtained
without resorting to gyrokinetics to order O(δ 2 fM 0 ).
If we had gyroaveraged C{fi } only to order δ, as most gyrokinetic models do, the
solution would have been simply
fi ' f M .
(70)
Substituting this solution into (55), we find the inconsistent result
µ
¶
Zcni
ZM c2 ni
Z
∇·
∇⊥ φ −
|∇⊥ φ|2 = ne − Zni −
∇2 pi .
(71)
2
BΩ
2Ti B
2M Ω2 ⊥
However, this quasineutrality equation is very different from the one we obtain by using
the full O(δ 2 fM 0 ) solution in (67), which simply gives
Zni = ne .
(72)
Therefore, the gyrokinetic quasineutrality equation reduces to the quasineutrality
condition when the exact O(δ 2 ) distribution function of (67) is employed. Equation (71)
is wrong because the O(δ) result of (70) is either inducing an O(δ 2 ) charge difference or
imposing the non-physical condition
µ
¶
Zcni
ZM c2 ni
Z
∇·
∇⊥ φ −
|∇⊥ φ|2 = −
∇2 p i .
(73)
2
BΩ
2Ti B
2M Ω2 ⊥
The difference between (71) and (72), given by (73), originates in O(δ 2 ) terms that
should have been cancelled by pieces of the distribution function of the same order.
The θ-pinch example illustrates the problem of using a lower order gyrokinetic
equation than needed, but it also highlights another issue. The potential does not
appear in the quasineutrality equation (72), and, therefore, it cannot be found using
it. This result is not surprising since in the simplified problem presented here the
dependence of fi with the axisymmetric potential appears only in the orders O(δ 3 ) and
O(δ 2 ν/Ω), as we will prove shortly. It is possible, of course, that if turbulence is included
in the problem, an axisymmetric zonal flow component of the potential appears in the
quasineutrality equation at O(δ 2 ), but in the absence of turbulence, the axisymmetric
component cannot be solved from quasineutrality even if the distribution function is
known to O(δ 2 ).
In the θ-pinch in absence of turbulence, the electrostatic potential is obtained from
a moment equation. The moment approach has the advantage of showing the effect of
Limitations of Gyrokinetics on Transport Time Scales
14
the potential in the ion density without having to calculate the distribution function
to higher order than O(δ 2 fM 0 ). The methodology we use here is presented for screw
pinches and dipolar configurations in [36]. Equation (72) is equivalent to ∇·J = 0, where
J is the current density. In the case of the axisymmetric θ-pinch, this is equivalent to
Jr = 0, where Jr is the radial component of the current density. In order to find Jr , we
use conservation of azimuthal angular momentum to get
↔
∇ · (cr π ·θ̂) = r(J × B) · θ̂,
(74)
↔
where θ̂ is the unit vector in the azimuthal direction and π is the ion viscosity, given by
µ
¶
Z
v2 ↔
↔
3
π= M d v fi vv −
I .
(75)
3
Since (J × B) · θ̂ = −BJr = 0, equation (74) reduces to r−1 ∂(r2 πrθ )/∂r = 0. In a case
without sources or sinks of momentum, the final equation for the potential is πrθ = 0.
Finding πrθ directly from the distribution function requires a higher order solution than
the one provided by the O(δ 2 ) gyrokinetic equation used so far. However, that problem
can be circumvented by using a moment approach, similar to the one in [37]. The
moment equation for the gyroviscosity is
↔
↔
↔
↔
Ω(π ×n̂ − n̂× π) =Kg + K⊥ ,
with
↔
µ
Kg = ∇ · M
and
↔
(76)
¶
Z
3
d v vvvfi
+ Zeni (∇φVi + Vi ∇φ)
(77)
Z
K⊥ = −M
d3 v vvC{fi }.
(78)
According to this relation,
πrθ = (Kg,rr − Kg,θθ + K⊥,rr − K⊥,θθ )/4Ω.
(79)
The gyrokinetic solution in (67) is high enough order to calculate πrθ by this moment
approach. The gyrokinetic variables R = r + R1 + R2 and E = E0 + E1 + E2 must be
Taylor expanded to get a second order distribution function dependent on the variables
r, v. Actually, it turns out that only the gyrophase dependent part of the distribution
function, fi − hfi i, is needed, where here the gyroaverage is done holding r, v|| and v⊥
fixed. This gyrophase dependent part is calculated in Appendix D, and the result is
(fi − hfi i)g = v · g⊥ − [vd · v + (v|| /4Ω)(v⊥ v × n̂ + v × n̂v⊥ ) : ∇n̂]B −1 ∂fi /∂µ0
↔
+ Ω−1 [(v|| n̂ + v⊥ /4)v × n̂ + v × n̂(v|| n̂ + v⊥ /4)] : h,
(80)
where the subindex g stands for non-collisional, and where
g⊥ = Ω−1 n̂ × [∇fi − (Ze/M )∇φ ∂fi /∂E0 ]
(81)
and
↔
h= ∇g⊥ − (Ze/M )∇φ ∂g⊥ /∂E0 .
(82)
Limitations of Gyrokinetics on Transport Time Scales
15
We also need to add the gyrophase dependent piece given by (38). For the Krook
operator it becomes
(fi − hfi i)c = −(ν/2Ω2 )fM 0 (M v 2 /Ti − 5)v⊥ · ∇ ln Ti .
(83)
When all these factors are taken into account, we find
µ
¶
∂ 1 2νpi ∂Ti
Kg,rr − Kg,θθ = −r
(84)
∂r r M Ω2 ∂r
and
·
µ
¶¸
µ
¶
νr ∂ 1 pi ∂Ti
νrpi ∂
c
∂φ
1 ∂pi
−
.
(85)
K⊥,rr − K⊥,θθ = −
+
Ω ∂r rB ∂r Zeni ∂r
Ω ∂r r M Ω ∂r
Using these results, πrθ = 0 gives
¶
µ
·
µ
¶¸
Z r
0
∂φ
1 ∂pi
∂
3 ∂
pi (r0 )U (r0 )
0 U (r )
0
= rB(r)
dr
c
+
ln B(r ) −
ln
, (86)
∂r Zeni ∂r
r0
∂r0
2 ∂r0
r0
0
where U = (2/M Ω)(∂Ti /∂r). Note the difference between this equation and (73). In
particular, notice that for an isothermal fM 0 , ∂Ti /∂r = 0, a radial Maxwell-Boltzmann
response is recovered from (86) as expected, but this is not a feature of the non-physical
forms (71) and (73).
5. Discussion
We have found an electrostatic gyrokinetic equation that provides the solution to O(δfM )
for both long and short wavelengths. Furthermore, the gyrokinetic variables are found
to high enough order to provide, at long wavelengths, the gyrophase dependent part of
the distribution function to O(δ 2 fM ) (allowing us to recover the gyroviscosity).
The gyrokinetic equation is complemented by a quasineutrality equation that
might be expected to provide the electrostatic potential in a self-consistent calculation.
However, we demonstrate that it is unable to retain the long wavelength components
of the potential if the distribution function is only exact to O(δfM ). The traditional
gyrokinetic approach is based on adjusting the potential each timestep according to
its effect on the gyromotion of the particles, while the gyrocenter motion is given by
the potential in the previous timestep. This procedure gives the right potential for
short wavelengths, on the order k⊥ ρ ∼ 1, since O(δ) accuracy is adequate, but fails as
longer and longer wavelengths are included in the analysis because their effect on the
gyromotion is averaged out and O(δ 2 ) and higher modifications to the Maxwellian are
required. It might seem that keeping more terms in the gyrokinetic equation to obtain a
higher order solution for the distribution function would be enough to find the potential,
but finding such a gyrokinetic equation for general geometry is difficult and its solution
by numerical means requires high numerical precision since terms smaller than O(δ 2 fM )
must be recovered without appreciable error to calculate the full axisymmetric potential
to lowest order.
The θ-pinch solution shows that the quasineutrality condition fails to provide a
solution in the steady state without turbulence even when a O(δ 2 fM ) solution is used.
Limitations of Gyrokinetics on Transport Time Scales
16
In the absence of a moment description, the axisymmetric potential can be only obtained
if the distribution function is known to O(δ 2 ν/Ω). This behavior is not exclusive to the
θ-pinch, as the same problem is found for up-down symmetric tokamaks [22], where
contributions to the distribution function of O(δ 3 ) and O(δ 2 ν/Ω) must be retained to
determine the axisymmetric radial electric field.
In our minds, the best solution to these problems is a combined kinetic and moment
approach that solves a gyrokinetic equation and a group of moment equations at the
same time. The potential will be given in this case not by a gyrokinetic quasineutrality
equation, but by ∇·J = 0. The calculation would be able to retain neoclassical viscosity
effects and the turbulent Reynolds stress that must be allowed to compete to determine
the potential. A moment approach usually has the advantage that it requires a lower
order distribution function [7]. A simplified example of this approach is the solution of
the θ-pinch presented in this paper, where the electrostatic potential is finally given by
the conservation of azimuthal angular momentum, which in turn is an integrated form
of ∇ · J = 0.
Acknowledgments
The authors are indebted to Jeffrey Freidberg of MIT for suggesting the use of the
θ-pinch as an illustrative example, to Bill Dorland of University of Maryland for his
support, and to Grigory Kagan of MIT for many helpful discussions.
This research was supported by the U.S. Department of Energy Grant No. DEFG02-91ER-54109 at the Plasma Science and Fusion Center of the Massachusetts
Institute of Technology and by the Center for Multiscale Plasma Dynamics of University
of Maryland.
Appendix A. First order gyrokinetic variables
In this appendix the detailed calculation of the gyrokinetic variables is carried out to first
order. It is convenient to express any term that contains the electrostatic potential φ in
gyrokinetic variables, mainly because we are not able to Taylor expand the electrostatic
potential components with k⊥ ρ ∼ 1. In order to do so, we will develop some useful
relations involving the potential φ in the first section of this appendix. With these
relations, the first order corrections, R1 , E1 , µ1 and ϕ1 , are derived.
Useful relations for φ. We first derive all possible gyrokinetic partial derivatives of φ
and their relation to one another. To do so, only R = r + Ω−1 v × n̂ + O(δ 2 L) is needed.
The derivative respect to the gyrocenter position is
∇R φ(r) = ∇φ + ∇R (r − R) · ∇φ = ∇φ + O(δT /eL) ' ∇φ.
(A.1)
The derivative respect to the energy is
∂φ/∂E = ∂(r − R)/∂E · ∇φ = O(δ 2 M/e) ' 0,
(A.2)
Limitations of Gyrokinetics on Transport Time Scales
17
since r − R only depends on E at O(δ 2 L).
√
Using r − R ∝ µ(ê1 sin ϕ − ê2 cos ϕ), the derivatives with respect to µ and ϕ can
be calculated to be
2
∂φ/∂µ = ∂(r − R)/∂µ · ∇φ ' −(M c/Zev⊥
)(v × n̂) · ∇φ
(A.3)
∂φ/∂ϕ = ∂(r − R)/∂ϕ · ∇φ ' −Ω−1 v⊥ · ∇φ.
(A.4)
and
We will need a more accurate relationship than (A.4) for the second order
corrections. It will be developed in Appendix B.
Calculation of R1 . The first order correction R1 is given by (12), where in this case,
Q0 = R0 = r. The total derivative of R0 is dR0 /dt = v = v|| n̂ + v⊥ , and its gyroaverage
gives hdR0 /dti = v|| n̂ + O(δvi ). By employing v⊥ = ∂(v × n̂)/∂ϕ0 , equation (12) gives
(29).
Calculation of E1 . The first order correction E1 is given by (12), where Q0 = E0 = v 2 /2
and dQ0 /dt = dE0 /dt = −(Ze/M )v · ∇φ. It is convenient to write E1 as a function of
R, E, µ and ϕ. To do so, we use (A.1) and (A.4) to find
−v · ∇φ = −v|| n̂ · ∇φ − v⊥ · ∇φ ' −v|| n̂ · ∇R φ + Ω ∂φ/∂ϕ.
(A.5)
Notice that n̂ · ∇R φe ¿ n̂ · ∇R φ because φe is smaller than φ. As a result, dE0 /dt =
−(Ze/M )v|| n̂·∇R φ+(ZeΩ/M )∂φ/∂ϕ and hdE0 /dti = −(Ze/M )v|| n̂·∇R φ, and equation
(12) gives (17).
Calculation of µ1 . Calculating µ1 requires more work than calculating any of the other
first order corrections since we want µ to be an adiabatic invariant to all orders of
interest. This requirement imposes two conditions to µ1 . One of them is similar to the
requirements already imposed to R1 and E1 , dµ0 /dt − Ω ∂µ1 /∂ϕ = hdµ0 /dti ≡ 0, but
there is an additional condition making µ0 + µ1 an adiabatic invariant to first order,
hd(µ0 + µ1 )/dti = O(δ 2 vi3 /BL).
(A.6)
The solution to both conditions is given by
¿
µ
˦
Z
1 ϕ 0 dµ0
dµ0
µ1 =
dϕ
−
+ hµ1 i.
Ω
dt
dt
(A.7)
Notice that the only difference with the result in (12) is that the gyrophase independent
term, hµ1 i, must be retained, making it possible to satisfy condition (A.6).
2
The total derivative for µ0 = v⊥
/(2B) is
2
/2B)v⊥ · ∇ ln B − (v||2 /B)n̂ · ∇n̂ · v⊥
dµ0 /dt = −(Ze/M B)v⊥ · ∇φ − (v⊥
− (v|| /2B)[v⊥ v⊥ − (v × n̂)(v × n̂)] : ∇n̂,
(A.8)
↔
2
where we have used the relations hv⊥ v⊥ i = (v⊥
/2)( I −n̂n̂) and
v⊥ v⊥ − hv⊥ v⊥ i = [v⊥ v⊥ − (v × n̂)(v × n̂)]/2.
(A.9)
Limitations of Gyrokinetics on Transport Time Scales
18
Notice that the gyrophase independent terms in (A.8) cancel exactly due to n̂ · ∇ ln B +
∇ · n̂ = 0, making µ0 an adiabatic invariant to zeroth order. The term that contains φ
in (A.8) is rewritten as a function of the gyrokinetic variables by using (A.4), to give
−(Ze/M B)v⊥ · ∇φ = (ZeΩ/M B)∂φ/∂ϕ.
Applying (A.7), µ1 is found to be given by (34). To get this result, we have employed
v⊥ = ∂(v × n̂)/∂ϕ0 and
∂[v⊥ (v × n̂) + (v × n̂)v⊥ ]/∂ϕ0 = 2[v⊥ v⊥ − (v × n̂)(v × n̂)].
(A.10)
2
/2BΩ)(n̂ · ∇ × n̂) was chosen to insure that condition
The average value hµ1 i = −(v|| v⊥
(A.6) is satisfied. In previous works [38, 20], it has been noticed that solving (A.6) may
be avoided and replaced by imposing the relation E = (dR/dt · n̂(R))2 /2 + µB(R) on
the gyrokinetic variables. This procedure works in this case, and allows us to find hµ1 i.
We have checked that this value satisfies condition (A.6), but this derivation is omitted
here because of its length.
Calculation of ϕ1 . The first order correction ϕ1 is given by (12), where Q0 = ϕ0 . The
zeroth order gyrophase ϕ0 is defined by v⊥ = v⊥ (ê1 cos ϕ0 + ê2 sin ϕ0 ), where v⊥ is the
magnitude of the perpendicular velocity and ê1 (r, t) and ê2 (r, t) are two orthonormal
vector fields constructed such that ê1 × ê2 = n̂. According to this definition, upon using
2
2
2
v⊥
∇v ϕ0 = −v × n̂ and v⊥
∇ϕ0 = v|| ∇n̂ · (v × n̂) + v⊥
∇ê2 · ê1 , the total derivative of ϕ0
is
e
− (v × n̂) · [∇ ln Ω + (v 2 /v 2 )n̂ · ∇n̂ − n̂ × ∇ê2 · ê1 ]
dϕ0 /dt = −Ω − (Z 2 e2 /M 2 c)∂ φ/∂µ
||
+
2
(v|| /2v⊥
)[v⊥ (v
⊥
× n̂) + (v × n̂)v⊥ ] : ∇n̂,
(A.11)
where we have used (A.9), and the potential φ(r, t) and the gyrofrequency Ω(r) have
been written as functions of the gyrokinetic variables by using (A.3) and Ω(r) '
Ω(R) + (r − R) · ∇Ω, respectively. The function Ω is given by
Ω = Ω(R) + (v|| /2)n̂ · ∇ × n̂ − v|| n̂ · ∇ê2 · ê1 + (Z 2 e2 /M 2 c)∂φ/∂µ.
(A.12)
Upon gyroaveraging (A.11) we obtain hdϕ0 /dti = −Ω + O(δ 2 Ω). Finally, ϕ1 is obtained
from (12) by employing v × n̂ = −∂v⊥ /∂ϕ0 and (A.10), giving
e
ϕ1 = −(Ze/M B)∂ Φ/∂µ
− Ω−1 v⊥ · [∇ ln B + (v 2 /v 2 )n̂ · ∇n̂ − n̂ × ∇ê2 · ê1 ]
||
−
2
)[v⊥ v⊥
(v|| /4Ωv⊥
⊥
− (v × n̂)(v × n̂)] : ∇n̂,
(A.13)
e is the function defined in (21). As a result, the total derivative of ϕ is
where Φ
dϕ/dt = hdϕ0 /dti + O(δ 2 Ω) = −Ω + O(δ 2 Ω).
Appendix B. Second order gyrokinetic variables
To construct the gyrokinetic variables to second order, higher order relations than the
ones developed in Appendix A are needed to express φ as a function of the gyrokinetic
variables. These extended relations are deduced in the first part of this appendix. Using
them, the second order corrections R2 and E2 are calculated. The magnetic moment
and the gyrophase are not required to higher order.
Limitations of Gyrokinetics on Transport Time Scales
19
More useful relations for φ. To calculate the second order corrections, v · ∇φ must be
expressed in gyrokinetic variables to order O(δT vi /eL). The total time derivative for φ
in r, v variables is
dφ/dt = ∂φ/∂t|r + v · ∇φ,
(B.1)
while as a function of the new gyrokinetic variables it becomes
dφ/dt = ∂φ/∂t|R, E, µ, ϕ + Ṙ · ∇R φ + Ė ∂φ/∂E + ϕ̇ ∂φ/∂ϕ.
(B.2)
Combining these equations gives an equation for v · ∇φ,
−v · ∇φ = (∂φ/∂t|r − ∂φ/∂t|R, E, µ, ϕ ) − Ė ∂φ/∂E − Ṙ · ∇R φ − ϕ̇ ∂φ/∂ϕ,
(B.3)
where the left side of the equation is of order O(T vi /eL). We analyze the right side term
by term. Noticing that φ(r, t) = φ(R+(r−R), t), the partial derivatives with respect to
time give the negligible contribution (∂φ/∂t|r − ∂φ/∂t|R, E, µ, ϕ ) = −∂(r − R)/∂t · ∇φ =
O(δ 2 T vi /eL), since the time derivative of r−R can only be of order O(ωδ 2 L) for a static
magnetic field. The partial derivative with respect to E is estimated in (A.2). Applying
it to our equation, we find that it is also negligible, Ė ∂φ/∂E = O(δ 2 T vi /eL). The
total derivative Ṙ has two different components, which we will calculate in detail in the
next paragraph. These components are the parallel velocity of the gyrocenter, v || n̂(R),
of order vi , and the drift velocity, vd , of order δvi . Using this information, we find
v || n̂(R) · ∇R φ = O(T vi /eL) and vd · ∇R φ = O(δT vi /eL). Finally, the last term in the
e
right side of (B.3) is ϕ̇ ∂φ/∂ϕ = O(T vi /eL), since ϕ̇ ∼ Ω and ∂φ/∂ϕ = ∂ φ/∂ϕ
∼ δT /e
according to (20). Neglecting all the terms smaller than O(δ) compared to v · ∇φ,
equation (B.3) becomes
e
−v · ∇φ = −v || n̂ · ∇R φ − vd · ∇R φ + Ω ∂ φ/∂ϕ.
(B.4)
Calculation of R2 . The second order correction R2 is given by (13), where Q0 = R0 = r
and Q1 = R1 = Ω−1 v × n̂. The total time derivative of R0 + R1 is
d(R0 + R1 )/dt = v|| n̂ − v · ∇(n̂/Ω) × v − (c/B)∇φ × n̂,
(B.5)
and its gyroaverage may be written as hd(R0 + R1 )/dti = v || n̂(R) + vd , where
2
v || = hv|| i + (v⊥
/2Ω)(n̂ · ∇ × n̂), and vd has been already defined in (23). The function
v || can be written as a function of the gyrokinetic variables. We express v|| as a function
of the lowest order gyrokinetic variables, expand about these lowest order gyrokinetic
variables, and insert R1 , µ1 and E1 to obtain
p
p
2
/2Ω)n̂ · ∇ × n̂
v|| = 2(E0 − µ0 B(r)) ' 2(E − µB(R)) − (v⊥
− (v|| /Ω)(v × n̂) · (n̂ · ∇n̂) − (4Ω)−1 [v⊥ (v × n̂) + (v × n̂)v⊥ ] : ∇n̂.
(B.6)
Finally, gyroaveragingp
and using hv⊥ (v × n̂) + (v × n̂)v⊥ i = 0 [a result that is deduced
from (A.9)] give v || = 2(E − µB(R)), which can be rewritten as (25).
Using (B.5) and (B.6), Taylor expanding n̂(r) about R and inserting the result into
(13) gives (30) and (31). To integrate, v × n̂ = −∂v⊥ /∂ϕ0 and (A.10) have been used.
Limitations of Gyrokinetics on Transport Time Scales
20
Calculation of E2 . Equation (13) gives E2 , where Q0 = E0 = v 2 /2 and Q1 = E1 =
e . The total derivative of E0 = v 2 /2 can be expressed as a function of the new
Zeφ/M
gyrokinetic variables to the requisite order by using (B.4) to obtain
e
dE0 /dt = −(Ze/M )v · ∇φ ' (Ze/M )[Ω ∂ φ/∂ϕ
− (v || n̂ + vd ) · ∇R φ].
(B.7)
e , use of gyrokinetic variables yields
From the definition of E1 = Zeφ/M
e
e
dE1 /dt = (Ze/M )[∂ φ/∂t
+ (v || n̂ + vd ) · ∇R φe − Ω ∂ φ/∂ϕ].
(B.8)
Adding both contributions together leaves
e
d(E0 + E1 )/dt = −(Ze/M )(v || n̂ + vd ) · ∇R φ + (Ze/M )∂ φ/∂t.
(B.9)
As a result, E2 is as shown in (18), and to this order, dE/dt is given by (22).
Appendix C. Jacobian of the gyrokinetic transformation
The inverse of the Jacobian is
¯
..
..
..
¯ ...
.
.
.
¯
¯
¯
∇R
∇E ∇µ ∇ϕ
¯
..
..
..
..
¯
.
1 ¯
.
.
.
=¯ .
.
.
..
¯ ..
..
..
J
.
¯
¯
∇v R
∇v E ∇v µ ∇v ϕ
¯
¯
..
..
..
.
¯
..
.
.
.
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯=¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
..
..
.. ¯¯
.
.
. ¯
¯
∇R
∇E ∇µ ∇ϕ ¯
..
..
.. ¯¯
..
.
.
.
. ¯
..
..
.. ¯¯ .
...
.
.
. ¯
0
∂E ∂µ ∂ϕ ¯¯
..
..
.. ¯¯
..
.
.
.
.
...
(C.1)
Employing that the terms in the left columns of the first form are to first approximation
↔
↔
∇R ' I and ∇v R ' Ω−1 I ×n̂, the determinant is simplified by combining linearly the
rows in the matrix to determine the second form, where
∂(. . .) = ∇v (. . .) − Ω−1 n̂ × ∇(. . .).
(C.2)
The second form of (C.1) can be simplified by noticing that the lower left piece of the
matrix is zero. Thus, the determinant may be written as
J −1 = det(∇R)[∂E · (∂µ × ∂ϕ)].
(C.3)
We analyze the two determinants on the right side independently. The matrix
↔
∇R is I +∇(Ω−1 v × n̂ + R2 ). Hence, det(∇R) = 1 + ∇ · (Ω−1 v × n̂ + R2 ). The
Jacobian must be obtained to first order only. The only important term to that order
in R2 is the term that contains the potential φ, since its gradient may be large, but
e × n̂] ' 0. Therefore, the determinant of ∇R becomes
∇ · R2 ' ∇ · [(c/BΩ)∇R Φ
det(∇R) = 1 − v · ∇ × (n̂/Ω).
(C.4)
For the second determinant in (C.3), we evaluate the columns of the matrix ∂E,
∂µ and ∂ϕ to the order of interest, using
∂E = v + ∇v E1 − Ω−1 n̂ × ∇E,
(C.5)
∂µ = v⊥ /B + ∇v µ1 − Ω−1 n̂ × ∇µ
(C.6)
Limitations of Gyrokinetics on Transport Time Scales
21
and
−2
∂ϕ = −v⊥
v × n̂ + ∇v ϕ1 − Ω−1 n̂ × ∇ϕ.
(C.7)
The determinant becomes
2
∂E · (∂µ × ∂ϕ) ' v|| /B(r) + n̂ · ∇v E1 /B + (∇v µ1 − Ω−1 n̂ × ∇µ) · [(v|| /v⊥
)v⊥ − n̂]
−(v|| /B)(∇v ϕ1 − Ω−1 n̂ × ∇ϕ) · (v × n̂).
(C.8)
From the definitions of E1 , µ1 and ϕ1 , we find their gradients in velocity space. We
e
need the gradients in velocity space of φe and ∂ Φ/∂µ.
The gradient ∇v φe is given by
e
e
e
∇v φe = ∇v E ∂ φ/∂E
+ ∇v µ ∂ φ/∂µ
+ ∇v ϕ ∂ φ/∂ϕ
+ ∇v R · ∇R φe
e
e
e
= B −1 v⊥ ∂ φ/∂µ
− v −2 v × n̂ ∂ φ/∂ϕ
+ Ω−1 n̂ × ∇R φ.
⊥
(C.9)
e
The gradient ∇v (∂ Φ/∂µ)
is found in a similar way. The gradients in real space are only
to be obtained to zeroth order. However, some terms of the first order quantities that
contain φ are important because they have steep gradients. Considering this, we find
e
∇E = (Ze/M )∇φ,
(C.10)
2
∇µ = −(v⊥
/2B)∇ ln B − (v|| /B)∇n̂ · v⊥ + (Ze/M B)∇φe
(C.11)
2
e
∇ϕ = (v|| /v⊥
)∇n̂ · (v × n̂) + ∇ê2 · ê1 − (Ze/M B)∇(∂ Φ/∂µ).
(C.12)
and
Due to the preceding considerations, equation (C.8) becomes
∂E · (∂µ × ∂ϕ) = [v || /B(r)](1 + Ω−1 v⊥ · ∇ × n̂),
(C.13)
where we have employed (B.6) to express v|| as a function of the gyrokinetic variables
2
and (v × n̂) · ∇n̂ · v⊥ − v⊥ · ∇n̂ · (v × n̂) = v⊥
n̂ · ∇ × n̂, a result that is deduced from
↔
2
v⊥
( I −n̂n̂) = v⊥ v⊥ + (v × n̂)(v × n̂).
(C.14)
Combining (C.4) and (C.13), the Jacobian of the transformation is found to be as given
by (44). Notice that to this order J = hJi as required by (40).
Finally, we prove that J satisfies the gyroaverage of (40) to the required order,
namely,
∂J/∂t + ∇R · {J[v || n̂(R) + vd ]} − (Ze/M )∂{J[v || n̂(R) + vd ] · ∇R φ}/∂E = 0,
(C.15)
where vd is given by (23). To first order, we obtain
J[v || n̂(R) + vd ] ' B(R) + (Bµ/v || Ω)n̂ × ∇R B + (Bv || /Ω)∇R × n̂ − (c/v || )∇R φ × n̂,(C.16)
where we have employedpn̂ × (n̂ · ∇R n̂) = (∇R × n̂)⊥ . Inserting (C.16) into (C.15) and
remembering that v || = 2(E − µB(R)) is enough to prove that (C.15), and thus (40)
gyroaveraged, are satisfied by the Jacobian to first order.
Limitations of Gyrokinetics on Transport Time Scales
22
Appendix D. Calculation of gyroviscosity at long wavelengths
Here we show how to obtain the gyroviscosity at long wavelengths (k⊥ ρ ¿ 1 and
k⊥ L ∼ 1). To simplify the calculation, the distribution function of ions will be assumed
to be Maxwellian to lowest order, i.e., fi (R, E, µ, t) ' f M (R, E, t), where f M is given
by (68). To recover the gyroviscosity, the ion distribution function must be written in
r, v variables. Taylor expanding fi to O(δ 2 f M ) gives
fi (R, E, µ, t) ' fi (r, E0 , µ0 , t) + (R1 + R2 ) · ∇fi + (E1 + E2 )∂fi /∂E0 + µ1 ∂fi /∂µ0
+ (1/2)R1 R1 : ∇∇f M + (1/2)E12 ∂ 2 f M /∂E02 + E1 R1 · ∇(∂f M /∂E0 ). (D.1)
Here, fi ' f M is used in the higher order terms and the contribution of the collisional
piece fe, given by (38), is intentionally ignored because its only effect is a small classical
transport contribution [8].
The gyroviscosity depends only on the gyrophase dependent part of fi and according
to equation (27) of [22] may be written as
Z
↔
π g = d3 v M vv (fi − hfi i) ,
(D.2)
where here the gyroaverage hfi i is now performed holding r, v|| and v⊥ fixed. Employing
(D.1), the gyrophase dependent part of fi is found to be
e + E2 ](∂fi /∂E0 )
fi − hfi i = (Ω−1 v × n̂ + R2 ) · ∇fi + [(Ze/M )(φe − hφi)
+ (4Ω2 )−1 [(v × n̂)(v × n̂) − v⊥ v⊥ ] : ∇∇f M + (Z 2 e2 /2M 2 )(φe2 − hφe2 i)∂ 2 f M /∂E02
e × n̂) − hφ(v
e × n̂)i] · ∇(∂f /∂E0 ) + (µ1 − hµ1 i)(∂fi /∂µ0 ),
+ (c/B)[φ(v
(D.3)
M
where we have used (A.9) to rewrite (v × n̂)(v × n̂) − h(v × n̂)(v × n̂)i.
The function φe must be written as a function of the r, v variables. To do so, first
we will write φ(r, t) as a function of the gyrokinetic variables by Taylor expansion to
O(δ 2 T /e) to find
φ(r, t) ' φ(R, t) − R1 · ∇R φ + (1/2)R1 R1 : ∇R ∇R φ − R2 · ∇R φ. (D.4)
The second term in the right side of the equation needs to be re-expanded in order to
express φ as a self-consistent function of the gyrokinetic variables to the right order.
The function R1 is, to O(δρ),
p
R1 = Ω−1 v × n̂ ' [ 2µB(R)/Ω(R)][ê1 (R) sin ϕ − ê2 (R) cos ϕ] − ∆ρ,
(D.5)
where the function ∆ρ(R, E, µ, ϕ, t) is O(δρ), but its exact form is not needed here.
Combining (D.4) and (D.5), φe is found to be
p
φe = φ − hφi ' −[ 2µB(R)/Ω(R)][ê1 (R) sin ϕ − ê2 (R) cos ϕ] · ∇R φ − R2 · ∇R φ
+ (∆ρ − h∆ρi) · ∇R φ + (4Ω2 )−1 [(v × n̂)(v × n̂) − v⊥ v⊥ ] : ∇R ∇R φ. (D.6)
We require φe as a function of r, v and t. Taylor expanding the first term in (D.6) gives
↔
2
/4Ω2 )( I −n̂n̂) : ∇∇φ
φe ' −Ω−1 (v × n̂) · ∇φ − h∆ρi · ∇φ − R2 · ∇φ − (v⊥
−(2Ω2 )−1 (v × n̂)(v × n̂) : ∇∇φ.
(D.7)
Limitations of Gyrokinetics on Transport Time Scales
23
This result allows us to rewrite some of the terms in (D.3) in a more convenient way.
For example, we obtain
e = −Ω−1 (v × n̂) · ∇φ − R2 · ∇φ + [v⊥ v⊥ − (v × n̂)(v × n̂)] : (∇∇φ)/4Ω2 . (D.8)
φe − hφi
For the higher order terms, we can simply use the lowest order result φe ' −Ω−1 (v ×
n̂) · ∇φ, which leads to
φe2 − hφe2 i = −(2Ω2 )−1 [v⊥ v⊥ − (v × n̂)(v × n̂)] : (∇φ∇φ)
(D.9)
and
e × n̂) − hφ(v
e × n̂)i = (2Ω)−1 ∇φ · [v⊥ v⊥ − (v × n̂)(v × n̂)].
φ(v
(D.10)
Using these expressions, the gyrophase dependent part of the distribution function
becomes
fi − hfi i = v · g⊥ + (µ1 − hµ1 i)∂fi /∂µ0 + R2 · G + E2 ∂fi /∂E0
+ (4Ω2 )−1 [(v × n̂)(v × n̂) − v⊥ v⊥ ] : [∇G − (Ze/M )∇φ ∂G/∂E0 ],
(D.11)
where g⊥ is given in (81) and
G = ∇fi − (Ze/M )∇φ ∂fi /∂E0 .
(D.12)
Thus, g⊥ = Ω−1 n̂×G. In the long wavelength limit, ∂/∂t ¿ vi /L, so E2 as given in (18)
is negligible since it contains a time derivative. Also, the zeroth order Fokker-Planck
equation for the ion distribution function is
v|| n̂ · G ≡ v|| n̂ · [∇fi − (Ze/M )∇φ ∂fi /∂E0 ] = C{fi } = 0,
(D.13)
since the ion distribution function is assumed to be Maxwellian to zeroth order. This
condition is important in (D.11) because it implies that the components of R2 that are
parallel to the magnetic field do not enter fi − hfi i. Therefore, employing the definition
e n̂ is negligible,
of R2 in (30) and using the fact that for long wavelenghts [c/(BΩ)]∇R Φ×
we obtain
R2 · G = Ω−1 [(v|| n̂ + v⊥ /4)v × n̂ + v × n̂(v|| n̂ + v⊥ /4)] : [∇(n̂/Ω) × G]
+ (v|| /Ω2 )v⊥ · ∇n̂ · G,
(D.14)
Equation (D.14) can be written in a more recognizable manner by using
↔
[n̂(v × n̂) + (v × n̂)n̂] : h = −Ω−1 n̂ · [∇G − (Ze/M )∇φ ∂G/∂E0 ] · v⊥
+ [n̂(v × n̂) + (v × n̂)n̂] : [∇(n̂/Ω) × G],
(D.15)
↔
where h is given in (82). The first term in the right side of (D.15) can be further
simplified by using (D.13) to obtain
n̂ · [∇G − (Ze/M )∇φ ∂G/∂E0 ] · v⊥ = v⊥ · ∇G · n̂ = −v⊥ · ∇n̂ · G.
(D.16)
As a result, (D.14) becomes
↔
R2 · G = (v|| /Ω)[n̂(v × n̂) + (v × n̂)n̂] : h
+ (4Ω)−1 [v⊥ (v × n̂) + (v × n̂)v⊥ ] : [∇(n̂/Ω) × G].
(D.17)
The gyrophase dependent part of the ion distribution function can now be explicitly
written as in (80). This is exactly the same gyrophase-dependent distribution function
found in [22]. Therefore, the same gyroviscosity as found there will be obtained.
Limitations of Gyrokinetics on Transport Time Scales
24
Appendix E. Quasineutrality equation at long wavelengths
In this Appendix we obtain the gyrokinetic quasineutrality equation at long wavelengths
(k⊥ ρ ¿ 1 and k⊥ L ∼ 1). To simplify the calculation, the ion distribution function is
assumed to be Maxwellian to lowest order. In quasineutrality, the ion distribution
function is again written in r, v variables, as has already been done in Appendix D, in
(D.1). The ion density is
Z
ni (r, t) = d3 v fi (R, E, µ, t).
(E.1)
This density can be calculated to O(δ 2 ni ) by using (D.1). Some of the terms are zero
R
because the integral over gyrophase, ϕ0 , is zero; for example, d3 v (R1 + R2 ) · ∇fi =
R
0 = d3 v E2 ∂fi /∂E0 . The ion density becomes
"
¶#
µ
Z
2
e ∂fi
e
Ze
φ
1
∂f
Ze
φ
∂
f
∂f
i
M
M
ni ' N̂i + d3 v
+
+
+ R1 · ∇
,
(E.2)
2
M ∂E0 B ∂µ0
2M ∂E0
∂E0
where N̂i (r, t) is the ion gyrocenter density, defined as the portion of the ion density
independent of φe and given by
Z
Z
Z
2
∂fi
1
3
3 v|| v⊥
N̂i = d v fi (r, E0 , µ0 , t) − d v
n̂ · ∇ × n̂
+ d3 v R1 R1 : ∇∇f M . (E.3)
2BΩ
∂µ0
2
The formula for N̂i can be simplified. The second term in the right side of the equation
R
2
is proportional to d3 v (v|| v⊥
/2B)∂fi /∂µ0 . This integral is simplified by changing to
2
2
the variables E0 = v /2, µ0 = v⊥
/2B and ϕ0 and integrating by parts,
Z
Z
2
XZ
∂fi
3 v|| v⊥ ∂fi
− dv
= −B
dE0 dµ0 dϕ0 σµ0
= d3 v v|| fi , (E.4)
2B ∂µ0
∂µ
0
σ
where σ = v|| /|v|| | is the sign of the parallel velocity, the summation in front of the
integral indicates that the integral must be done for both signs of v|| , and we have used
the equality d3 v = dE0 dµ0 dϕ0 B/|v|| |. The third term in the right side of (E.3) is
proportional to
Z
↔
M d3 v (v × n̂)(v × n̂) : ∇∇f M = ( I −n̂n̂) : ∇∇pi .
(E.5)
The final function N̂i reduces to the result shown in (56).
In (E.2), φe appears in several integrals. We perform these integrals by integrating
R
e i /∂E0 +
first in the gyroangle to simplify the expressions. For the integral d3 v φ(∂f
R
2π
B −1 ∂fi /∂µ0 ), only the gyrophase integral 0 φe dϕ0 is needed, but φe must be known as
a function of the variables r, v to O(δ 2 T /e) to be consistent with the order of the Taylor
expansion. This has already been done in Appendix D and the result is given in (D.7),
leading to
Z 2π
v2 ↔
1
φe dϕ0 ' −h∆ρi · ∇φ − ⊥2 ( I −n̂n̂) : ∇∇φ.
(E.6)
2π 0
2Ω
Limitations of Gyrokinetics on Transport Time Scales
25
with ∆ρ the difference shown in (D.5). The function ∆ρ is found by Taylor expanding
to O(δ 2 L), giving
2
∆ρ = −(2Ω2 )−1 (v × n̂)(v × n̂) · ∇ ln B + (M c/Zev⊥
)µ1 (v × n̂) + Ω−1 ϕ1 v⊥
+ (v⊥ /Ω2 )(v × n̂) · (sin ϕ0 ∇ê1 − cos ϕ0 ∇ê2 ).
(E.7)
To simplify this equation, the gradients of the unit vectors ê1 and ê2 are expressed in the
alternate forms ∇ê1 = −(∇n̂ · ê1 )n̂ − (∇ê2 · ê1 )ê2 and ∇ê2 = −(∇n̂ · ê2 )n̂ + (∇ê2 · ê1 )ê1 ,
giving
2
∆ρ = −(2Ω2 )−1 (v × n̂)(v × n̂) · ∇ ln B + (M c/Zev⊥
)µ1 (v × n̂) + Ω−1 ϕ1 v⊥
− Ω−2 (v × n̂) · ∇n̂ · (v × n̂)n̂ − Ω−2 v⊥ (v × n̂) · ∇ê2 · ê1 ,
(E.8)
and its gyroaverage
2
2
h∆ρi = −(v⊥
/Ω2 )∇⊥ ln B − (v||2 /Ω2 )n̂ · ∇n̂ − (c/BΩ)∇⊥ φ − (v⊥
/2Ω2 )(∇ · n̂)n̂,
(E.9)
where we have used
2
4
2 2
hµ1 (v × n̂)i = −(cv⊥
/2B 2 )∇⊥ φ − (v⊥
/4BΩ)∇⊥ ln B − (v⊥
v|| /2BΩ)n̂ · ∇n̂
(E.10)
and
2
2
hϕ1 v⊥ i = −(c/2B)∇⊥ φ − (v⊥
/2Ω)∇⊥ ln B − (v||2 /2Ω)n̂ · ∇n̂ + (v⊥
/2Ω)n̂ × ∇ê2 · ê1 .(E.11)
These expressions are found by using the definitions of µ1 and ϕ1 , given by (34)
and (A.13), and employing the lowest order expressions φe ' −Ω−1 (v × n̂) · ∇φ,
R
2
e
e = ϕ φe dϕ0 ' Ω−1 v⊥ · ∇φ and ∂ Φ/∂µ
' (M c/Zev⊥
)v⊥ · ∇φ.
Φ
Substituting (E.9) in (E.6) gives
µ
¶ µ
¶
Z 2π
2
2
v
1
1
1
v
c
⊥
⊥
2
φe dϕ0 = − ∇ ·
∇
φ
+
v
−
n̂
·
∇n̂
·
∇φ
+
|∇⊥ φ|2 , (E.12)
⊥
||
2
2
2π 0
2
Ω
2 Ω
BΩ
where we have used
↔
∇ · (Ω−2 ∇⊥ φ) = Ω−2 ( I −n̂n̂) : ∇∇φ − 2Ω−2 ∇ ln B · ∇⊥ φ − Ω−2 n̂ · ∇n̂ · ∇φ
− Ω−2 (n̂ · ∇φ)∇ · n̂.
(E.13)
Note that the gyroaverage of φe is O(δ 2 T /e), which means that the integral is O(δ 2 ni ),
and the lowest order distribution function, f M , can be used to write ∂fi /∂E0 '
−(M/Ti )f M and ∂fi /∂µ0 ' 0. All these simplifications lead to the final result
µ
¶
Z
³ c
´ M c2 n
∂fi
1 ∂fi
Ze
i
3 e
d vφ
+
= ni ∇ ·
∇⊥ φ −
|∇⊥ φ|2 , (E.14)
2
M
∂E0 B ∂µ0
BΩ
Ti B
The other two integrals in (E.2) can be done by using the lowest order results
φe ' −Ω−1 (v × n̂) · ∇φ and fi ' f M . The integrals are
Z
2 2
2
M c2 n i
3 Z e e2 ∂ f M
dv
φ
=
|∇⊥ φ|2
(E.15)
2
2
2
2M
∂E0
2Ti B
and
Z
ce
× n̂) · ∇
d v φ(v
B
3
µ
∂f M
∂E0
¶
=
c
∇ni · ∇⊥ φ.
BΩ
(E.16)
Limitations of Gyrokinetics on Transport Time Scales
Using (E.14), (E.15) and (E.16), (E.2) becomes
³ cn
´ M c2 n
i
i
ni = N̂i + ∇ ·
∇⊥ φ −
|∇⊥ φ|2 ,
BΩ
2Ti B 2
26
(E.17)
where N̂i is given by (56). Therefore, the quasineutrality condition is as shown in (55).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
Dorland W, Jenko F, Kotschenreuther M and Rogers B N 2000 Phys. Rev. Lett. 85 5579
Jenko F, Dorland W, Kotschenreuther M and Rogers B N 2000 Physics of Plasmas 7 1904
Rosenbluth M N and Hinton F L 1998 Phys. Rev. Lett. 80 724
Hinton F L and Rosenbluth M N 1999 Plasma Phys. Control. Fusion 41 A653
Xiao Y, Catto P J and Molvig K 2007 Physics of Plasmas 14 032302
Xiao Y, Catto P J and Dorland W 2007 Physics of Plasmas 14 055910
Catto P J and Simakov A N 2005 Physics of Plasmas 12 012501
Catto P J and Simakov A N 2005 Physics of Plasmas 12 114503, and Catto P J and Simakov A N
2005 Physics of Plasmas 13 129901
Wong S K and Chan V S 2007 Physics of Plasmas 14 112505
Rosenbluth M N, Rutherford P H, Taylor J B, Frieman E A and Kovrizhnikh L M 1971 Plasma
Phys. Controlled Nucl. Fusion Res. 1 495
Wong S K and Chan V S 2005 Physics of Plasmas 12 092513
Hinton F L and Wong S K 1985 Physics of Fluids 28 3082
Connor J W, Cowley S C, Hastie R J and Pan L R 1987 Plasma Phys. Control. Fusion 29 919
Catto P J, Bernstein I B and Tessarotto M 1987 Physics of Fluids 30 2784
Newton S L, Helander P and Catto P J 2007 Physics of Plasmas 14 062301
Lee X S, Myra J R and Catto P J 1983 Physics of Fluids 26 223
Bernstein I B and Catto P J 1985 Physics of Fluids 28 1342
Catto P J 1977 Plasma Physics 20 719
Hitchcock D A and Hazeltine R D 1978 Plasma Physics 20 1241
Catto P J, Tang W M and Baldwin D E 1981 Plasma Physics 23 639
Hazeltine R D 1973 Plasma Physics 15 77
Simakov A N and Catto P J 2005 Physics of Plasmas 12 021105
Lee W W 1983 Physics of Fluids 26 556
Lee W W 1987 Journal of Computational Physics 72 243
Ross D W and Dorland W 2002 Physics of Plasmas 9 5031
Candy J and Waltz R E 2003 Phys. Rev. Lett. 91 045001
Hinton F L and Hazeltine R D 1976 Rev. Mod. Phys. 48 239
Candy J and Waltz R E 2003 Journal of Computational Physics 186 545
Heikkinen J A, Kiviniemi T P, Kurki-Suonio T, Peeters A G and Sipilä S K 2001 Journal of
Computational Physics 173 527
Heikkinen J A, Janhunen J S, Kiviniemi T P and Kåll P 2004 Contrib. Plasma Phys. 44 13
Grandgirard V et al 2006 Journal of Computational Physics 217 395
Dubin D H E, Krommes J A, Oberman C and Lee W W 1983 Physics of Fluids 26 3524
Hahm T S, Lee W W and Brizard A 1988 Physics of Fluids 31 1940
Brizard A 1989 J. Plasma Physics 41 541
Qin H, Cohen R H, Nevins W M and Xu X Q 2007 Physics of Plasmas 14 056110
Simakov A N and Catto P J 2006 Physics of Plasmas 13 112509
Catto P J and Simakov A N 2004 Physics of Plasmas 11 90
Northrop T G and Rome J A 1978 Physics of Fluids 21 384